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Full text of "Treatise On Analysis Vol-Ii"

18   CANONICAL DECOMPOSITIONS OF A MEASURE         209
\fd\fi\ =1 JnfM- Since /t = h  LU| with |/i(an)| = 1 for all n (13.16.3), we

w= 1                                                                                                                           oo

have /x(<jD{fln}) = pn with |/?J = yn, and f/rfju =  Ai/(*n) for every /z-integ-

J                   n= 1

rable function/.

Conversely, consider an arbitrary denumerable sequence (an) of points of
X, and a sequence (yn) of real numbers >0 such that  yn < + 00 for all

aneK

compact subsets K of X. We have already seen (13.1.3) that

/^v(/) = f ynf(an)
= i

is a positive measure on X. With the same notation as above, it is clear that
v = sup((pDn  v), and since <pDn - v is concentrated on Dn and hence on D, it

n

follows that v is concentrated on D. Also we have v({an}) = yn. For if K is a
compact neighborhood of an, then for each e > 0 there exists an integer m such
that ]T yp <^ e. If /: X - [0, 1] is a continuous mapping which takes

peK,pm

the value 1 at an, and the value 0 on X  K and at points ak such that k ^ n
and k <a m (4.5.2), then we have v({an}) g v(/) <; yn H- s; on the other hand
we may choose the neighborhood K such that v(K) ^ v({an}) + e (13.7.9), and
a fortiori yn ^ v(/) ^ v({an}) + e. Since s was arbitrary, our assertion is
proved.

PROBLEMS

1.    Let p, a be two atomic measures on a locally compact space X, and let M, N be the
smallest sets carrying |p|, |cr , respectively. Show that p, cr are disjoint if and only if
M r\ N = 0. Hence give an example of an atomic measure v on the interval I = [0, 1]
of R such that I is the support (13.19) of v+ and of v".

2.   (a)   Let ^ be a positive measure on a locally compact space X. If A c X is universally
measurable and not ja-negligible, show that there exists a positive measure v carried by
A such that v ^ 0 and v <i p,. (Observe that A contains a compact set K which is not
/x-negligible.)

(b)    Let M be a universally measurable subset of X. For a positive measure /x to be
carried by M it is necessary and sufficient that v should be disjoint from every positive
measure carried by X  M (use (a)).

(c)    If M is closed in X, show that the vector subspace of M(X) consisting of the
measures carried by M is vaguely closed in M(X).

(d)    Show that Lebesgue measure on 1= [0,1] is the vague limit of a sequence of
atomic measures carried by a fixed denumerable subset D of I. The subspace of M(I)
consisting of measures carried by D is therefore not vaguely closed.le set fr)l fr ^1  e fri* ^i], then we have